On a class of mathematical ecosystems with random jumps

In order to model random population dynamics with cycles in ecosystems, we construct a diffusion process with random jumps from the boundary. The evolution of this new process is investigated. It is shown that the behavior of the whole process is uniquely determined by its first hitting time and the transition before jumps. The stationary distribution is also given and it is applied to predict the long-run population density of stochastic logistic growth models with jumps.